Tangential dimensions II. Measures

Notions of (pointwise) tangential dimension are considered, for measures of R-N. Under regularity conditions (volume doubling), the upper resp. lower tangential dimension at a point x of a measure p can be defined as the supremum, resp. infimum, of local dimensions of the measures tangent to mu at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a measure, namely which is able to detect the "oscillations" of the dimension at a given point, even when the local dimension exists, namely local upper and lower dimensions coincide. These definitions are tested on a class of fractals, which we call translation fractals, where they can be explicitly calculated for the canonical limit measure. In these cases the tangential dimensions of the limit measure coincide with the metric tangential dimensions of the fractal defined in [7], and they are constant, i.e. do not depend on the point. However, upper and lower dimensions may differ. Moreover, on these fractals, these quantities coincide with their noncommutative analogues, defined in previous papers [5, 6), in the framework of Alain Connes' noncommutative geometry.